Controllability Gramian

In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system

$\dot{x} = A x + B u$

the controllability Gramian is given by

$W_c (t) = \int\limits_0^t e^{A\tau} B B^T e^{A^T \tau} d\tau = \int\limits_0^t e^{A(t-\tau)} B B^T e^{A^T(t-\tau)} d\tau$

If the matrix W_c is nonsingular, i.e. W_c has full rank, for any t > 0, then the pair (A,B) is controllable.

Linear time-variant state space models of form

$\dot{x}(t) = A(t) x(t) + B(t) u(t)$,

y(t) = C(t)x(t) + D(t)u(t)

are controllable in an interval [t₀,t₁] if the rows of the matrix productΦ(t₀,τ)B(τ) where Φ is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of Φ(t₀,τ)B(τ). To have linear independency Gramian matrix W_c have to be nonsingular, i.e., invertible.

$W_c(t) = \int\limits_{t_0}^{t_1} \Phi(t_0,\tau)B(\tau)B^T(\tau)\Phi^T(t_0,\tau) d\tau$

See also

• Observability Gramian
• Gramian matrix

References

External links

• Controllability Gramian Lecture notes to ECE 521 Modern Systems Theory by Professor A. Manitius, ECE Department, George Mason University.